1
vote
2answers
37 views

Check my proof: Big O notation

I was asked the following: We are given the functions $f(n)=n^{10\log(n)}$ and $g(n)=(\log (n))^n$. Which of the following statements is true: $f(n)\in\mathcal{O}(g(n))$, $f(n) \in ...
0
votes
2answers
50 views

Does the definition range remains the same?

In solving this inequality (transcribed from here) $$\left(\frac23\right)^{\log_{0.5}(x^2+4x+4)}<\left(\frac94\right)^{\log_2(x^2-3x-10)}$$ we eventually reach the point where $ ...
0
votes
1answer
25 views

How to define this logarithmic function

I am trying to get my head around the definition of this function (that I concocted as an exercise in defining a function). Let $f$ denote the function satisfying: $f(0) = +\infty$, and $f(+\infty) = ...
1
vote
1answer
29 views

How can I read logarithmic scale?

I've got this histograms: How can I read that logarithmic scale? For example, on the histogram 1 there is approximately $10^{-3}$ value at y-axis at 2 value at x-axis. Does it meant that there is a ...
1
vote
1answer
30 views

Domain of definition of the function

I was going through some questions of Relations and Functions and now I am stuck to one. Question says Question: Domain of definition of the function $$f(x)=\frac{9}{9-x^2}+\log_{10}(x^3-x)$$ ...
1
vote
1answer
22 views

If $g(x) = \text{arctanh}\ (\log x)$, find $g'(x)$.

If $g(x) = \text{arctanh}\ (\log x)$, find $g'(x)$. I tried to separate the terms first and I got $\dfrac12 (\log(1+\log x) - \log(1-\log x))$. The answer is $\dfrac1{x(1-\log x)^2}$.
0
votes
1answer
22 views

Function that represents growth with specific specs

I'm looking for a function that represents growth based on the following specs: In a range of 365 days the function may grow from (almost) zero to a maximum of 1. It should have some kind of ...
0
votes
1answer
14 views

Applications of logarithmic functions in shapes and geometries

As I understand this logarithmic functions are a family of functions where the equation for $f(x)$ is written like so $$\begin{align} f(x) = & \log_{a} x \\ & \mathtt{where\ }a\mathtt{\ is\ ...
1
vote
2answers
222 views

Find a function that satisfies the condition.

Let $\epsilon > 0$ be fixed and $t$ a variable that takes values in the universal covering space of ${\mathbb{C} \setminus \{0\}}$. Find a continuous function $f(s$) such that $$|t \log t| = |t| ...
1
vote
1answer
50 views

Rewriting in $y=A_0\cdot e^{at}$

How do you rewrite $y = −8(1.589)^{t − 3}$ in $y=A_0e^{at}$ form for appropriate constants $A_0$ and $a?$ For other problems I took the $\ln$ of the number inside the parenthesis. So for example I ...
1
vote
2answers
29 views

Function application (word problem)

The problem: My work so far: $3=log(\frac{A}{A_0})$--->$10^3=\frac{A}{A_0}$ $\frac{A}{A_0}=1000$ (Am I done there?) Plugging it in: $M=log(\frac{1900000}{1000})$ $10^M = \frac{1900000}{1000}$ ...
-2
votes
1answer
33 views

How to solve for $k$ when the area about the $x$ axis and under the graph of the $f(x) = \frac1x$ from interval $x = [2, k]$ is equal to $\ln(4)$?

What approach would be ideal in solving for a number $k$ when the area about the $x$ axis and under the graph of the function $f(x) = \frac1x$ from interval $x = [2, k]$ is equal to $\ln(4)$?
0
votes
2answers
24 views

Find the inverse of the function

Find the inverse of the function $f(x) = -2 \cdot4^{2(x-3)} - 1$.
1
vote
1answer
32 views

Function inverse mapping [0, +inf) to [0, 1)

I have a measure ($x$) which the domain is $[0, +\infty)$ and measure some sort of variability. I want to create a new measure ($y$) that represents regularity and is inverse related to $x$. It is ...
0
votes
1answer
35 views

Prove this logarithm equation

I keep getting the wrong answer. Can someone please correct my working out a^x=b^(1-x) In(a)^x=In(b)^(1-x) xIn(a)=(1-x)In(b) xIn(a)=In(e)-xIn(b) xIn(a)+xIn(b)=In(e) x[In(a)+In(b)]=Ine ...
0
votes
1answer
45 views

Inverse Function of Logarithm

The answer is A but I don't understand why! $ -2 \log_e (x^2) $ can be re-written as $ -4 \log_e(x) $ right? but why do these two graphs look different? the graph $-2 \log_e (x^2) $ is one to ...
0
votes
1answer
17 views

Maximum value of constant in logarithm problem

The first thing I did was: make: (x-1)^2 - k > 0 (x-1)^2 > k don't know what to do after this point... the maximum value of k is 9 i dont really understand what the maximum value of k is? ...
0
votes
1answer
380 views

Graphing: Given two points on a graph, find the logarithmic function that passes through both.

Is there such a method to do this? I would like to come up with a logarithmic function (a graph that looks like a square root graph) that passes through two given points. Haven't had any luck in ...
0
votes
0answers
37 views

Scaling a big range of small numbers to a small range of big numbers

I'm trying to make a volume meter in a Flash program. I have data coming in like: 0.008 0.0005 0.1 0.02 These numbers indicate the volume of a sound coming in ...
0
votes
2answers
196 views

Find Log equation from data points

I have the following data points, (left hand column goes from 0-127, right hand column goes from 30-22000 hz. Is there any calculator I can use to find a "log" function of this data, so that it comes ...
0
votes
1answer
64 views

Steps to Graph Exponential Equations & Absolute Value

how to sketch: $-e^{|-x-1|} + 2$ Can someone clarify: $|f(x)|:$ we draw $f(x)$ and then reflect the ($-y$ parts) in the $x$-axis $f|(x)|:$ we draw $f(x)$ and then reflect the ($-x$ parts) in the ...
1
vote
1answer
30 views

Show that $g(x)=x\ln{x}$ and $g(x)=e^x$ are bounded below.

Show that $g(x)$ is bounded below, for $0\leq x$: a) $g(x) = \left\{ \begin{array}{ll} 0 & \mbox{if } x=0 \\ x\ln{x} & \mbox{if } x>0 \end{array} \right.$ b) $g(x)=e^x$ For (a), ...
0
votes
1answer
15 views

Creating a constrained log function

Good morning, I have a series of values that I intend to use as the exponents and I would like to create a log function so that: $Log_x(y_1)=.1$ $Log_x(y_2)=.2$ $Log_x(y_3)=.3$ ... ...
0
votes
1answer
71 views

Give the domain and range of $y=\log(x-3)+2$

I am so confused. I think the domain is $x>3$ but is the range ARN or is it $y>0$ . . .
10
votes
2answers
266 views

$\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$

Prove that: $\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$, for $n > 1,\, n\in \mathbb{N}$ For example. For $n=2$, we have $\lfloor ...
2
votes
1answer
25 views

What is the number of real roots of $(\log x)^2- \lfloor\log x\rfloor-2=0$ $\lfloor\,\cdot\,\rfloor$ represents the greatest integer

Question : What is the number of real roots of $(\log x)^2- \lfloor\log x\rfloor-2=0$. $\lfloor\,\cdot\,\rfloor$ represents the greatest integer function less than or equal to $x$. I know how to ...
1
vote
2answers
44 views

Need help with a proof concerning zero-free holomorphic functions.

Suppose $f(z)$ is holomorphic and zero-free in a simply connected domain, and that $\exists g(z)$ for which $f(z) =$ exp$(g(z))$. The question I am answering is the following: Let $t\neq 0$ be a ...
1
vote
1answer
63 views

Creating a function with logarithmic growth

I have some knobs with an internal value of $0$ to $1$. These represent a value in a range, like $1$ to $1000$. Case in point, I would like to be able to change the scale/growth of the display value. ...
2
votes
4answers
87 views

graphing $f(x)=x \ln \left(1+\frac{1}{x}\right)$

I was assigned to draw the graph of this function $f(x)$=$x\ln(1+{1\over x})$. And when I calculate $\lim_{x\to \infty} f(x)$ I get $1$ but the teacher it's not correct even though its graph on the ...
0
votes
1answer
68 views

For $0<a<b$, show $1-\dfrac{a}{b} < \log\left( \dfrac{b}{a} \right) < \dfrac{b}{a}-1$

Prove that if, $0 < a < b$ Then $1-\dfrac{a}{b} < \log\left( \dfrac{b}{a} \right) < \dfrac{b}{a}-1$
2
votes
1answer
52 views

Show that $\operatorname{ln}(n!)=\Theta(n\operatorname{ln}(n))$

Another question about asymptotic approximations. We are asked to show that $\operatorname{ln}(n!)=\Theta(n\operatorname{ln}(n))$ I'm stuck tho and can use help. What I did is: ...
0
votes
0answers
33 views

check my short simple proof - Functions are of same magnitude. Asymptotic notation.

A simple question with a short solution I thought of, but I would like verification. $f(n)$ is a function that approaches infinity as $n$ approaches infinity. We are asked to show that ...
0
votes
1answer
33 views

Allowed values for $x$ in $\log_2(x)$

$$y=\log_2x$$ What are the allowed values for $x$ in this function? How do I calculate it? (I know how it works for normal functions with fractions and other stuff, but this one I'm stuck)
0
votes
2answers
2k views

Writing an equation for a log function given the graph

I have the following graph for a logarithmic function $f$: I don't know any thing about writing an equation for a logarithmic function by knowing it's graph. All what I know is how to draw a graph ...
0
votes
1answer
40 views

Minimum value of $ f(x) = x\log_2x +(1-x)\log_2(1-x) $ [closed]

What is the minimum value of the following function for $ 0<x<1 $ ? Here the base of logarithm is 2 . $ f(x) = x\log_2x +(1-x)\log_2(1-x) $
2
votes
1answer
90 views

Why isn't $\log(-1)$ defined?

Why isn't $\log(-1)$ defined. It can be defined as being equal to $i\pi$. Why don't we define the $log$ function over Complex Numbers as well.
0
votes
1answer
45 views

function symmetric around a point

I need some quick help solving this: What is y(ln(2))if the function y satisfies $$\frac{dy}{dx}=1-y^2$$ and is symmetric about the point (ln(4),0)? I know that a function is symmetric about the ...
0
votes
2answers
53 views

Use a graph to estimate the time at which the number was increasing most rapidly

For the period from 2000 to 2008, the percentage of households in a certain country with at least one DVD player has been modeled by the function $f(t) = \frac{87.5}{1 + 17.1e^{−0.91t}}$ where the ...
1
vote
1answer
45 views

At which parameter value $c>0$ do the number of solutions of $\log(1+x^2)=x^c$ change?

I'm looking at the functions $x\mapsto \log(1+x^2)$ and $x\mapsto x^c,\ c>0$ on the interval $\mathbb R^+_0$. I'm interested in the properties of $$\log(1+x^2)=x^c.$$ Graphically, for small $c$, ...
0
votes
1answer
112 views

Expressing logarithms as ratios of natural logarithms

$$\frac{\log_2 x}{\log_3 x}=\frac{\ln x}{\ln 2} \div \frac{\ln x}{\ln3}$$ Why can logarithms be written as ratios of natural logarithms? Can you explain it abstractly, please? Example of an ...
1
vote
1answer
52 views

Understanding a question on iterated logarithms

I have in front of me a math problem that I do not understand. That's to say, I don't understand what is being asked of me. Problem: We can define $\log_2**(x) = log_2*(log_2*(x))$ and the function ...
2
votes
1answer
40 views

Making logarithmic function go higher

I am looking at logarithmic functions, and, lets say, log2 (x+3) is having a bit of a growth rate between 0-10 values of ...
2
votes
1answer
76 views

Online logarithm drawing

I am looking for a site that will give me the output of my logarithms. What I want to do, is I want to input, in example log(2), and I want it to draw an output ...
1
vote
1answer
52 views

domain of function with logarithmic terms.

what will be the domain of function given below? $$y=1+3(\log(\sin(x))+\log(\csc(x)))$$ in book it is given this is valid for the values of angles of 1st and 2nd quadrant only. why this function is ...
1
vote
2answers
54 views

Find the inverse function…

So, I have the function $$f(x)=\frac{2^x-2^{-x}}{2}.$$ I tried finding the inverse function the usual way I do, but I guess I'm stuck with these degrees. So far, I've come to this form ...
1
vote
2answers
93 views

Filling in 'x' in a log function

if $3^5=x$ (exponential equation) converts to log form gives $log_3x=5$ that makes sense. $$ 3^5 = 243 \Rightarrow x=243 $$ So if I take the log form again: $log_3x=5$ and replace $x$ with $243$. I ...
4
votes
5answers
185 views

How do I solve such logarithm

I understand that $\log_b n = x \iff b^x = n$ But all examples I see is with values that I naturally know how to calculate (like $2^x = 8, x=3$) What if I don't? For example, how do I solve for $x$ ...
2
votes
1answer
42 views

determining sign of function containing logarithm.

I would like to know the sign of the following term in general. I tried approximately and it was negative. Is there any $m_0$ such that for all $n>m>m_0$, the following function is positive or ...
1
vote
0answers
36 views

What's the most straight forward way to show that a function is increasing?

I am trying to show that: $$\frac{2}{n}\log\Gamma\left(\frac{x}{2}\right) - \log\Gamma\left(\frac{x+n-1}{n}\right)$$ is an increasing function for $x \ge 5$ and $n > 2$ One way to do this would ...
1
vote
1answer
237 views

Comparing rates of change: which function increases faster?

I am comparing two functions for $x \ge 1$: $$f(x) = \ln(\lfloor\frac{x}{9}\rfloor!) - \ln(\lfloor\frac{x}{10}\rfloor!) - \ln(\lfloor\frac{x}{90}\rfloor!)$$ $$g(x) = (2.07766)\sqrt{\frac{x}{9}} + ...